The least distance of distinct vision of a person is `75` cm. The focal length of the reading spectacles for such a person should be

Options

A. `37.5` cm
B. `40` cm
C. `25` cm
D. `50` cm

Explanation

To determine the focal length of the reading spectacles for a person whose least distance of distinct vision is 75 cm, we can follow these steps:

### Step 1: Understand the least distance of distinct vision
The least distance of distinct vision (D) for a normal eye is typically 25 cm. In this case, the person's least distance of distinct vision is 75 cm. This means that the person can only see objects clearly when they are at least 75 cm away.

### Step 2: Identify the requirements for the spectacles
To enable the person to see objects clearly at the normal least distance of distinct vision (25 cm), we need to use a lens that will allow objects at this distance to be focused at the person's least distance of distinct vision (75 cm).

### Step 3: Use the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Where:
- \( f \) is the focal length of the lens,
- \( v \) is the image distance (which will be -75 cm since the image is virtual and on the same side as the object),
- \( u \) is the object distance (which will be -25 cm since the object is placed at the least distance of distinct vision).

### Step 4: Substitute the values into the lens formula
Substituting the values into the lens formula:
\[ \frac{1}{f} = \frac{1}{-75} - \frac{1}{-25} \]

### Step 5: Simplify the equation
Calculating the right side:
\[ \frac{1}{f} = -\frac{1}{75} + \frac{1}{25} \]
Finding a common denominator (which is 75):
\[ \frac{1}{f} = -\frac{1}{75} + \frac{3}{75} = \frac{2}{75} \]

### Step 6: Calculate the focal length
Now, taking the reciprocal to find \( f \):
\[ f = \frac{75}{2} = 37.5 \text{ cm} \]
Since the lens is used to correct vision, it will be a concave lens, which means the focal length will be negative:
\[ f = -37.5 \text{ cm} \]

### Conclusion
The focal length of the reading spectacles for this person should be \( -37.5 \) cm.

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